The line x cos `alpha +y sin alpha =p` is tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1.`if

Options

A. `a^(2) cos ^(2) alpha-b^(2)sin^(2)alpha =p^(2)`
B. `a^(2) sin ^(2) alpha+b^(2)cos^(2)alpha =p^(2)`
C. `a^(2) cos ^(2) alpha+b^(2)sin ^(2)alpha =p^(2)`
D. `a^(2)cos^(2)alpha+b^(2) sin^(2) alpha=p`

Explanation

To determine the condition under which the line \( x \cos \alpha + y \sin \alpha = p \) is tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Understand the equations
We have the equation of the ellipse given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
And the equation of the line given by:
\[ x \cos \alpha + y \sin \alpha = p \]

### Step 2: Use the parametric equation of the tangent to the ellipse
The parametric equation of the tangent to the ellipse at an angle \( \theta \) is given by:
\[ \frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} = 1 \]
We will compare this with the line equation.

### Step 3: Compare the coefficients
From the line equation \( x \cos \alpha + y \sin \alpha = p \), we can rewrite it in the form:
\[ \frac{x \cos \alpha}{p} + \frac{y \sin \alpha}{p} = 1 \]
Now, we can compare the coefficients:
- Coefficient of \( x \): \( \frac{\cos \theta}{a} = \frac{\cos \alpha}{p} \)
- Coefficient of \( y \): \( \frac{\sin \theta}{b} = \frac{\sin \alpha}{p} \)

### Step 4: Solve for \( \cos \theta \) and \( \sin \theta \)
From the first equation:
\[ \cos \theta = \frac{a \cos \alpha}{p} \]
From the second equation:
\[ \sin \theta = \frac{b \sin \alpha}{p} \]

### Step 5: Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \)
Substituting the expressions for \( \cos \theta \) and \( \sin \theta \) into the identity:
\[ \left(\frac{a \cos \alpha}{p}\right)^2 + \left(\frac{b \sin \alpha}{p}\right)^2 = 1 \]

### Step 6: Simplify the equation
This leads to:
\[ \frac{a^2 \cos^2 \alpha}{p^2} + \frac{b^2 \sin^2 \alpha}{p^2} = 1 \]
Multiplying through by \( p^2 \):
\[ a^2 \cos^2 \alpha + b^2 \sin^2 \alpha = p^2 \]

### Conclusion
Thus, the condition for the line to be tangent to the ellipse is:
\[ a^2 \cos^2 \alpha + b^2 \sin^2 \alpha = p^2 \]